Those notes are in line with a path entitled "Symplectic Geometry and Geometric Quantization" taught through Alan Weinstein on the collage of California, Berkeley (fall 1992) and on the Centre Emile Borel (spring 1994). the single prerequisite for the direction wanted is a data of the elemental notions from the idea of differentiable manifolds (differential varieties, vector fields, transversality, etc.). the purpose is to offer scholars an advent to the guidelines of microlocal research and the similar symplectic geometry, with an emphasis at the position those principles play in formalizing the transition among the maths of classical dynamics (hamiltonian flows on symplectic manifolds) and quantum mechanics (unitary flows on Hilbert spaces). those notes are supposed to functionality as a advisor to the literature. The authors confer with different assets for plenty of information which are passed over and will be bypassed on a primary examining.

This booklet is the 6th variation of the vintage areas of continuous Curvature, first released in 1967, with the former (fifth) variation released in 1984. It illustrates the excessive measure of interaction among crew idea and geometry. The reader will enjoy the very concise remedies of riemannian and pseudo-riemannian manifolds and their curvatures, of the illustration idea of finite teams, and of symptoms of modern growth in discrete subgroups of Lie teams.

Thus, energy levels of the 1-dimensional harmonic oscillator corresponding to level sets for which a particular value of is admissible are given by E = n . As one can read in any textbook on quantum mechanics, the actual quantum energy levels are E = (n + 1/2) . The additional 1/2 can be explained geometrically in terms of the non-projectability of the classical energy level curves, as we shall soon see. 39 Prequantum bundles and contact manifolds Prequantizability can be described geometrically in terms of principal T bundles with connection over T ∗ M .

We then turn to the semi-classical approximation and its geometric counterpart in this new context, setting the stage for the quantization problem in the next chapter. 2). 32 The classical picture The hamiltonian description of classical motions in a configuration space M begins with the classical phase space T ∗ M . A riemannian metric g = (gij ) on M induces an inner product on the fibers of the cotangent bundle T ∗ M , and a “kinetic energy” function which in local coordinates (q, p) is given by kM (q, p) = 1 2 g ij (q) pi pj , i,j where g ij is the inverse matrix to gij .